Thoughts about math, modeling, music, midlife, Montclair, and occasional things not beginning with the letter M.

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At the start of October, I went to see Elvis Costello at Town Hall, a small theater near Times Square with a history of hosting political meetings. You can see framed programs from some of those meetings, hazy, fading memories of the past, as you walk into the building:

The show had been loosely billed as a review of Elvis’s life and career, aligned with the memoir he released last summer, and I was looking forward to a few hours spent listening to him resculpting his songs in new and deeply satisfying ways. (I still felt warm from the time I saw him carry this off several years ago, at Carnegie Hall of all places, when he opened the show by playing the entire first side of My Aim is True, sang songs from all over the map, and told the kind of stories you might hear from the cool uncle you wish you had.)

For the first half hour, the show felt much like that night at Carnegie Hall. The stage was dominated by a giant TV, which showed 80’s Elvis videos as we waited for him to come on, then, once he did, ran a slide show of old family photos (hazy snapshots from the past again) and pictures from his early tours behind him as he played (solo, with acoustic guitar). He gave us both old and recent standards, culminating in a lovely rearrangement of Everyday I Write the Book. But as I burrowed deeper into my plush chair, getting comfortable for a couple more hours with Elvis’s fabulous songbook, he walked stage right, sat down at the piano, and smashed the time machine to bits.

In a halting, almost shattered voice, accompanied by the most austere, jagged piano figures you could imagine, he sang Shipbuilding, perhaps his greatest song. Written at the time of the Falklands war, Shipbuilding is a biography of out-of-work men in the English shipyards, waiting for the war to restart the shipyards and bring them jobs,

A new winter coat and shoes for the wifeAnd a bicycle on the boy’s birthday

and also for the warships that emerge from the shipyards to take those boys away, and bring them to their graves. “Is it worth it,” Elvis asks, and then asks again in a later verse that I hadn’t known before,

A small bunch of flowers is all that you get And a box to bury the baby

I had heard Elvis sing the song before, but not like this. I wish I knew how to describe the cocktail of regret and loss and compromise and inevitability I heard in his voice, but all I can do is point you to another song: if you listen to Springsteen’s Highway Patrolman, you’ll catch the same emotions when you hear Bruce’s voice trail off as he sings When it’s your brother, sometimes you look the other way at the end of the second verse. Hearing Elvis now, you could see shipyards, but also car factories on the outskirts of town, and mom and pop stores on Main Street, all of them filled with people’s dread of decisions they might know they’ll come to regret but still can’t avoid making. By the time the song’s resolution — diving for dear life, when we should be diving for pearls — exploded from the stage, you felt the full weight of the present day, of this election. Somehow, in just four minutes, Elvis had managed to make you feel the dilemmas of Trump voters deep down in your bones, more convincingly than any analysis I’ve read before or since the election. But he was only getting started.

He sang Deep Dark Truthful Mirror, an answer record in some ways, about facing up to the consequences of whatever choices you’ve made. Then, still at the piano, he started to tell us about a musical he was working on with the playwright Sarah Ruhl. It is an adaptation of the 1957 film A Face in the Crowd, which tells the story of the media-driven rise and fall of an American demagogue. If hearing Elvis sing Shipbuilding wasn’t enough to convince you he was addressing today’s political climate, the new songs he had written for the musical made it even plainer. He sang as the anti-hero, promising the angry and ignored to be their champion. He sang silly jingles that scoot back and forth across your screen and sell you silly things. And he sang for and with the faces in the crowd, all of us looking ourselves in the mirror and trying to find some common ground. The songs were theatrical, raw, maybe still works in progress — but they felt necessary and Elvis got them across. I am no fan of musicals, but I can’t wait to see what these songs will be like, and what they will mean, by the time they hit the stage, in the full blossoming of Trump’s America.

And then the finale. Given the range of Elvis’s songbook, it’s ironic that the song he is most associated with in the public imagination, What’s So Funny ‘Bout Peace, Love, and Understanding, is one he didn’t even write. But it is, and you won’t feel like you got your money’s worth unless you hear him play it, will you? Still, I’m not sure the song ever felt like it had more at stake than when Elvis sang it at Town Hall to close this show. And, as he sang it, the giant TV behind him projected a late 70’s, Armed Forces-era poster with the headline of the evening, Elvis’s answer to the America of Donald J. Trump, the next president of the United States: DON’T JOIN!

Soon it was over, and it was hard to believe we actually had to leave the building. But we did, and as I walked downstairs, the faded programs on the walls, from Town Meetings of long ago, were suddenly as sharp and timely as anything you might see on TV or on your Facebook feed. The past is not dead, not even past, as Faulkner said, and now all of us have many more town meetings to go to.

Modeling this election was never about win probabilities (i.e., saying that Clinton is 98% likely to win, or 71% likely to win, or whatever). It was about finding a way to convey meaningful information about uncertainty and about what could happen. And, despite the not-so-great headline, this article by Nate Silver does a pretty impressive job.

First, let’s have a look at what not to do. This article by Sam Wang (Princeton Election Consortium) explains how you end up with a win probability of 98-99% for Clinton. First, he aggregates the state polls, and figures that if they’re right on average, then Clinton wins easily (with over 300 electoral votes I believe). Then he looks for a way to model the uncertainty. He asks, reasonably: what happens if the polls are all off by a given amount? And he answers the question, again reasonably: if Trump overperforms his polls by 2.6%, the election becomes a toss-up. If he overperforms by more, he’s likely to win.

But then you have to ask: how much could the polls be off by? And this is where Wang goes horribly wrong.

The uncertainty here is virtually impossible to model statistically. US presidential elections don’t happen that often, so there’s not much direct history, plus the challenges of polling are changing dramatically as fewer and fewer people are reachable via listed phone numbers. Wang does say that in the last three elections, the polls have been off by 1.3% (Bush 2004), 1.2% (Obama 2008), and 2.3% (Obama 2012). So polls being off by 2.6% doesn’t seem crazy at all.

For some inexplicable reason, however, Wang ignores what is right in front of his nose, picks a tiny standard error parameter out of the air, plugs it into his model, and basically says: well, the polls are very unlikely to be off by very much, so Clinton is 98-99% likely to win.

Always be wary of models, especially models of human behavior, that give probabilities of 98-99%. Always ask yourself: am I anywhere near 98-99% sure that my model is complete and accurate? If not, STOP, cross out your probabilities because they are meaningless, and start again.

How do you come up with a meaningful forecast, though? Once you accept that there’s genuine uncertainty in the most important parameter in your model, and that trying to assign a probability is likely to range from meaningless to flat-out wrong, how do you proceed?

Well, let’s look at what Silver does in this article. Instead of trying to estimate the volatility as Wang does (and as Silver also does on the front page of his web site, people just can’t help themselves), he gives a careful analysis of some possible specific scenarios. What are some good scenarios to pick? Well, maybe we should look at recent cases of when nationwide polls have been off. OK, can you think of any good examples? Hmm, I don’t know, maybe…

Aiiieeee!!!!

Look at the numbers in that Sun cover. Brexit (Leave) won by 4%, while the polls before the election were essentially tied, with Remain perhaps enjoying a slight lead. That’s a polling error of at least 4%. And the US poll numbers are very clear: if Trump overperforms his polls by 4%, he wins easily.

In financial modeling, where you often don’t have enough relevant history to build a good probabilistic model, this technique — pick some scenarios that seem important, play them through your model, and look at the outcomes — is called stress testing. Silver’s article does a really, really good job of it. He doesn’t pretend to know what’s going to happen (we can’t all be Michael Moore, you know), but he plays out the possibilities, makes the risks transparent, and puts you in a position to evaluate them. That is how you’re supposed to analyze situations with inherent uncertainty. And with the inherent uncertainty in our world increasing, to say the least, it’s a way of thinking that we all better start becoming really familiar with.

The models were plain as day. What the numbers were telling us was that if the polls were right, Clinton would win easily, but if they were underestimating Trump’s support by anywhere near a Brexit-like margin, Trump would win easily. Shouldn’t that have been the headline? Wouldn’t you have liked to have known that? Isn’t it way more informative than saying that Clinton is 98% or 71% likely to win based on some parameter someone plucked out of thin air?

I went to bed around 11 when the outcome seemed obvious, woke up around 2 and couldn’t resist making sure it was really true, then lay awake in bed for the next few hours trying to process it. I can’t remember another time when I found it so hard to even catch hold of a coherent thought. Here are my three best attempts.

A lot of us are saying this isn’t the country we thought we knew. Look, here’s what you thought you knew a day ago: if you picked 100 people across the country at random, about 49 would vote for Hillary, about 46 for Trump, the remaining 5 or so for someone else, and things would trundle along as usual. Now it turns out that one or two of those people that you thought would vote for Hillary actually voted for Trump. That’s a big deal, and it has huge consequences, but it’s not like you were wrong about all, or even most, of those people, your fellow Americans. This is still the same country that voted for Obama twice. You were just wrong about one or two people out of 100, and anyway you don’t know what they were thinking.

One thing that *is* different is that the world looks a lot more uncertain now than many of us are used to. So let’s stop saying, like we know for sure, that Trump won because of racism, or misogyny, or stupidity, or whatever you’re absolutely sure is the reason. You were absolutely sure yesterday that Hillary would win, and look what happened. If we’re entering a very different world, let’s not reduce it to easy judgments, or leave the responsibility to figure out that world to other people.

Let’s not forget too that the level of anxiety that many of us are feeling now is closer to the level of anxiety that many other people feel, and cope with gracefully, every day. Yes it sucks to lose your Certainty Privilege, but we’ve probably been unusually lucky to have had it up until now. Right now I am coping with my own uncertainty by hanging on to the things that I am genuinely certain of, like how my sons, age 9 and age 7, deserve better than Trump. Going to try to make this a better world for them. Onward.

I have an 8:00 meeting this morning, so I’m going to take the early (6:36) bus in. I almost never go in this early, so to be safe, I am already downstairs putting on my shoes at 6:29. Sometimes I joke that I live so close to the bus stop that if I hear the bus from the second floor window of my house, I still catch it as it goes by. That’s an exaggeration, but not by much. Now I hear a sound outside that could be the bus, but I know it’s not, it’s never this early. Still, I peek out the door.

It’s the bus, already on the corner. Crap.

I dash out. The bus stop is diagonally across from my house, and fortunately, the bus is waiting at a red light. This gives me a chance: I run down to the corner and across the street with the light. Now the bus and I are facing each other, I wave, cross the other way when the light turns as the bus waits an extra couple seconds, and get on. I am feeling good about the world and my place in it.

Still, 6:29, that’s super-early. Unless maybe this is the previous bus, running super-late? I think I should say something to the driver. “Excuse me, which bus on the schedule is this?” It’s the morning, everything is rushed, who knows if I actually said “Excuse me.”

The bus driver, not one I’m familiar with, is not pleased. He barks at me: “It’s whatever bus it is when I get here!” I look behind him, see the bus is mostly empty. I know from experience that if it were the previous bus, running late, it would be overcrowded, so this must really be the 6:36. I give the driver my ticket and try to explain: you’re running way too early, people who count on this bus will miss it, then the next buses are overcrowded and run late, etc. I ask him to wait a few minutes to get back on schedule. He remains very hot, says (yells) that I better stop talking to him, threatens to call the police if I don’t get away from him. The good thing about him yelling at me is that the bus is still not moving. A couple people, breathless from running down the street, get on the bus while we’re going back and forth. I count this as a small victory.

A large man walks up the aisle from the back of the bus. I am hoping for a little support from a fellow commuter. “You better fucking stop talking to him so he can go!” he yells at me. “You’re holding the bus up! He’s going to call the cops!” This guy looks like he’d like nothing better than to slug me. “He’s calling the cops!” The bus driver is not calling the cops. My fellow passenger, though he is louder than the driver, does not actually take a swing at me. I sit down in one of the many empty seats.

The bus doesn’t move.

I take my book out of my bag and try to disappear into it. It is eerily silent. In moments of suspense, time feels suspended. The bus still doesn’t move.

Finally, after what must have been only a short wait but didn’t feel that way, the bus huffs and trundles forward. I check the time. It is exactly 6:36. We have been waiting for just a few minutes.

I mull things over as the bus heads into the city. I feel good that I managed to stay calm through the whole episode. Maybe this will be a good story to tell the children — the importance of keeping your cool. It occurs to me that the bus driver ended up doing exactly what I asked him to — waiting to leave until he was scheduled to, to the minute. Or was that just a coincidence? Something makes me decide to talk to him again, though I can’t tell exactly how or why I make that decision.

The bus pulls in to Port Authority. I am going to wait for everyone else to get off and then try to approach the driver. As the people ahead of me are getting off, I hear the usual end-of-trip courtesies: passengers say thank you, driver says you’re welcome or have a good day. That’s encouraging: this isn’t one of those guys who’s so nasty that people give up on saying thanks at the end of the ride.

Once is everyone else is off, I get off too. The driver is standing to the side. “Thanks for waiting a few minutes at the bus stop,” I say to him. “I wasn’t trying to pick a fight, I just…”

The driver is looking me in the eye. “Yeah, I know,” he says. “I get what you were saying.” I talk to him about the schedule, he says he understands, sometimes they get off schedule and it’s hard to keep track. We are talking to each other like people now. He says he’s sorry, and he seems totally sincere. I tell him my name, because that seems like a talking-like-people thing to do, and he tells me his. I was hoping to make peace, but this is more than I expected; in under a minute, our conversation has turned comfortably fraternal. I am feeling good about the world again as I head inside the bus terminal.

Now another man slides over to me. He is fiftyish, tall, thin, with slightly graying hair, wearing jeans and a blazer. “You know,” he says,”I saw the bus driver yelling at you, I got a video on my phone. I’m going to call the bus company.”

I don’t actually want him to call the bus company. I tell him that I just had a good talk with the driver, and that he seemed very direct and sincere. My companion isn’t impressed. “He’s just afraid for his job,” he says. He tells me that the driver has done this (I think he means run ahead of schedule) once before, that people were running for the bus and he didn’t care. He looks at me a little conspiratorially. “I think this guy just hates white people.”

Well. The driver, like the majority of bus drivers I encounter, is black. I am white. The guy I’m talking to now is white. The passenger who was yelling at me earlier is white. Most people on the bus are white, with a few exceptions. What is the logic here, how do you decide that a black man driving a bus full of mostly white people hates them because they’re white? Does the white man who came up to yell at me before hate white people too? Why do we look differently at angry black people than at angry white people?

The man I’m talking to is going his way and I’m going mine, and there isn’t time to ask these questions. But I as I walk up 8th Avenue on a sunny end-of-summer morning, I realize that I’ve lost the tidy story I was going to tell my kids about the virtues of keeping calm. And perhaps a bit of my confidence in those virtues as well.

So here you are on Amazon’s web page, reading about Cathy O’Neil’s new book, Weapons of Math Destruction. Amazon hopes you buy the book (and so do I, it’s great!). But Amazon also hopes it can sell you some other books while you’re here. That’s why, in a prominent place on the page, you see a section entitled:

Customers Who Bought This Item Also Bought

This section is Amazon’s way of using what it knows — which book you’re looking at, and sales data collected across all its customers — to recommend other books that you might be interested in. It’s a very simple, and successful, example of a predictive model: data goes in, some computation happens, a prediction comes out. What makes this a good model? Here are a few things:

It uses relevant input data.The goal is to get people to buy books, and the input to the model is what books people buy. You can’t expect to get much more relevant than that.

It’s transparent. You know exactly why the site is showing you these particular books, and if the system recommends a book you didn’t expect, you have a pretty good idea why. That means you can make an informed decision about whether or not to trust the recommendation.

There’s a clear measure of success and an embedded feedback mechanism. Amazon wants to sell books. The model succeeds if people click on the books they’re shown, and, ultimately, if they buy more books, both of which are easy to measure. If clicks on or sales of related items go down, Amazon will know, and can investigate and adjust the model accordingly.

Weapons of Math Destruction reviews, in an accessible, non-technical way, what makes models effective — or not. The emphasis, as you might guess from the title, is on models with problems. The book highlights many important ideas; here are just a few:

Models are more than just math. Take a look at Amazon’s model above: while there are calculations (simple ones) embedded, it’s people who decide what data to use, how to use it, and how to measure success. Math is not a final arbiter, but a tool to express, in a scalable (i.e., computable) way, the values that people explicitly decide to emphasize. Cathy says that “models are opinions expressed in mathematics” (or computer code). She highlights that when we evaluate teachers based on students’ test scores, or assess someone’s insurability as a driver based on their credit record, we are expressing opinions: that a successful teacher should boost test scores, or that responsible bill-payers are more likely to be responsible drivers.

Replacing what you really care about with what you can easily get your hands on can get you in trouble. In Amazon’s recommendation model, we want to predict book sales, and we can use book sales as inputs; that’s a good thing. But what if you can’t directly measure what you’re interested in? In the early 1980’s, the magazine US News wanted to report on college quality. Unable to measure quality directly, the magazine built a model based on proxies, primarily outward markers of success, like selectivity and alumni giving. Predictably, college administrators, eager to boost their ratings, focused on these markers rather than on education quality itself. For example, to boost selectivity, they encouraged more students, even unqualified ones, to apply. This is an example of gaming the model.

Historical data is stuck in the past. Typically, predictive models use past history to predict future behavior. This can be problematic when part of the intention of the model is to break with the past. To take a very simple example, imagine that Cathy is about to publish a sequel to Weapons of Math Destruction. If Amazon uses only purchase data, the Customers Who Bought This Also Bought list would completely miss the connection between the original and the sequel. This means that if we don’t want the future to look just like the past, our models need to use more than just history as inputs. A chapter about predictive models in hiring is largely devoted to this idea. A company may think that its past, subjective hiring system overlooks qualified candidates, but if it replaces the HR department with a model that sifts through resumes based only on the records of past hires, it may just be codifying (pun intended) past practice. A related idea is that, in this case, rather than adding objectivity, the model becomes a shield that hides discrimination. This takes us back to Models are more than just math and also leads to the next point:

Transparency matters! If a book you didn’t expect shows up on The Customers Who Bought This Also Bought list, it’s pretty easy for Amazon to check if it really belongs there. The model is pretty easy to understand and audit, which builds confidence and also decreases the likelihood that it gets used to obfuscate. An example of a very different story is the value added model for teachers, which evaluates teachers through their students’ standardized test scores. Among its other drawbacks, this model is especially opaque in practice, both because of its complexity and because many implementations are built by outsiders. Models need to be openly assessed for effectiveness, and when teachers receive bad scores without knowing why, or when a single teacher’s score fluctuates dramatically from year to year without explanation, it’s hard to have any faith in the process.

Models don’t just measure reality, but sometimes amplify it, or create their own. Put another way, models of human behavior create feedback loops, often becoming self-fulfilling prophecies. There are many examples of this in the book, especially focusing on how models can amplify economic inequality. To take one example, a company in the center of town might notice that workers with longer commutes tend to turn over more frequently, and adjust its hiring model to focus on job candidates who can afford to live in town. This makes it easier for wealthier candidates to find jobs than poorer ones, and perpetuates a cycle of inequality. There are many other examples: predictive policing, prison sentences based on recidivism, e-scores for credit. Cathy talks about a trade-off between efficiency and fairness, and, as you can again guess from the title, argues for fairness as an explicit value in modeling.

Weapons of Math Destruction is not a math book, and it is not investigative journalism. It is short — you can read it in an afternoon — and it doesn’t have time or space for either detailed data analysis (there are no formulas or graphs) or complete histories of the models she considers. Instead, Cathy sketches out the models quickly, perhaps with an individual anecdote or two thrown in, so she can get to the main point — getting people, especially non-technical people, used to questioning models. As more and more aspects of our lives fall under the purview of automated data analysis, that’s a hugely important undertaking.

You can measure how irresistible the Bangles are by the number of days (now two and counting since I saw them play live) that they’ve had me humming a song I had been pretty sure I hated.

Their show at Irving Plaza in New York on Saturday night really got cooking about half an hour in, with a lovely rendition of Big Star’s September Gurls. No matter what you think of the Bangles, you can’t deny the fact that they are the greatest cover band ever, and it’s not particularly close.

Next they played several tunes, at once primal and shiny, from Ladies and Gentlemen… The Bangles!, a just-released, and excellent, compilation of early recordings. “From when we were baby Bangles,” said one of the ladies, and my inability to remember which one was speaking only testifies to the band’s egalitarianism — everybody sings, everybody harmonizes, everybody talks. Then came Hazy Shade of Winter, a Simon and Garfunkel song once upon a time, but the Bangles took possession of it long ago, with blistering guitar lines and harmonies well beyond the original (!), and they’re not giving it back. Another highlight, kicking off the encore, was How is the Air Up There, which mixed garage rock with punk with a jangling and joyous charm that is entirely the Bangles’ own. When you hear the band on record, they sometimes suffer from 80’s-style overproduction, but all the songs they played at Irving Plaza were reduced to their essence, and the band got all them across. I am not a fan of recording shows on video, but I couldn’t help filming a bit of this one, just to have a keepsake.

The band closed the show, inevitably I suppose, with Eternal Flame, their biggest hit and the Freebird of their universe, more or less. I assumed this would be a letdown, especially after the vitality of the main set: despite Susanna Hoffs’s committed vocal, Eternal Flame always struck me as overproduced mush, pure pop cheese. But there’s a reason people yell and flick their lighters and refuse to go home till they hear Freebird: everybody likes a sing-along. After an hour and a half of speeding down Bangle freeway together, it was time for band and audience to meld into a single whole, and the ladies were smart and confident enough to make that happen by slowing down the pace and ending with a group hug of a ballad. So drummer Debbi Peterson came out from behind her kit and joined her bandmates on guitar, polka dots of light swirled through the audience, the cheesy lyrics became a call to harmony inside the club (Close your eyes! Give me your hand!), and there was no way you could keep yourself from singing along. It was corny, genuinely warm, and fully earned, a glorious way to send a thousand people jammed tightly in a small space out into the New York night. Maybe the song is treacle, but now I can’t stop humming it.

I was still feeling the thrill of the Bangles show two days later when I remembered how another band had recently made me feel almost exactly the same way. It was Sleater-Kinney, the Bangles’ spiritual successors in more ways than one, who came back last year after an almost ten year hiatus, playing just as fiercely as ever. When I went to see them, they too had ripped through an hour and a half of take-no-prisoners rock and roll… and then slowed things down and ended the show with their ironic ballad, Modern Girl. Only they played it as a sing-along, audience and band joined into one just like they would at the Bangles’ show, and the song completely transcended its irony. When we all sang the chorus together — My whole life is like a picture of a sunny day! — everyone knew the line as written was a deeply cutting takedown of consumer culture, but at the same time, in that room, there was no way you could help feeling it really was a sunny day, with this band back in your life. The moment was lovely, warm, musical, joyous, and more than a little Bangle-esque.

Maybe the best thing about the Pythagorean theorem is how it puts math and non-math people on a pretty equal footing. We all know what it says (right triangle, squares of sides, hypotenuse), we all agree it’s Important Math with a capital M… and most of us don’t have much idea, if any, why it’s true. Seriously. Ask another math person if you don’t believe me. If you’re lucky, they might point you to a picture that looks more or less like this:

This is sometimes called a proof without words, but here are a few words to guide you, just in case. We’ve got the usual notation: a and b are sides of a right triangle, c is the hypotenuse. We build a super-square with side length a+b and break it up in two ways. On the right, we divide it into one (white) sub-square with side length a (area a2), another with side b (area b2), and four (colored) copies of the triangle. On the left, we rearrange the four triangles so that their complement is a new white sub-square with side c (area c2). White area on the left = white area on the right, so c2 = a2 + b2.

Lovely as this is, it feels like a nifty conjuring trick. The Pythagorean equation is the most direct thing in the world, a2 + b2 = c2, and the best we can do is to rearrange triangles inside a big square? Surely there must be a way to cut up c2 into a2 and b2 directly.

Well, there is! I first ran across what I’m about to show you a few weeks ago, loved it, and was surprised that (1) I hadn’t seen it before and (2) it doesn’t seem to be widely known, though the idea actually goes back to Euclid. Perhaps you’ll feel the same way once you see it. Here goes:

The set-up. What’s so special about right triangles? Well, one thing is that they have an amazing self similarity property. Draw a line segment out from the vertex with the right angle toward the hypotenuse and perpendicular to it. It divides our original right triangle up into two smaller ones:

Let’s stick to the same notation we had before. Let c be the hypotenuse of our original right triangle, running along the bottom. Let a and b be the sides, a the one on the left, b the one on the right. Then a is the hypotenuse of the green right triangle on the left, and b is the hypotenuse of the blue right triangle on the right. Obviously (foreshadowing), the areas of the two smaller triangles add up to the area of the original.

The self similarity property is that the two smaller right triangles are both similar to the original one! In other words, all three triangles have the same three interior angles, which means that you can rotate and scale each one into any of the others. Put simply, all three triangles have the same shape. Can you see why? Let’s compare those interior angles: the big triangle has a right angle of 90 degrees, and two others, which we will call (say the one on the left, in the green triangle) and (on the right, in the blue triangle). The key point is that the angles of any triangle have to add up to 180 degrees, so two angles of a triangle always determine the third. The green triangle has an angle of on the left that it inherits from the original big triangle, and a 90 degree angle in the middle, so its third angle, at the top, must be . (Essentially: the green triangle and the big triangle have two angles in common, so they must have all three in common.) Similarly, the blue triangle has an angle of on the right that it inherits from the original triangle, and a 90 degree angle in the middle (again, two angles in common), so its third angle must be . Same angles, same shape.

The pay-off. Stare for a minute at those three similar triangles, with the areas of the two smaller ones adding up to the area of the bigger one. Wouldn’t it be great if the triangle with hypotenuse a had area a2, the one with hypotenuse b had area b2, and the one with hypotenuse c had area c2? It’s not true, of course. But it’s almost true! If you read my last post, you know that in fact the area of a right triangle with hypotenuse c and interior angle is

.

The green and blue triangle have exactly the same interior angles as the big one. So their areas are given by the same exact formula, with c replaced by a and b, respectively. The areas have to add up, so we have:

.

Now just divide out the common factor of , and you’re left with

.

What just happened? The way Pythagoras’s equation fell out of equating areas may seem like a bit of a magic trick too, but it’s actually based on the very fundamental idea of scale invariance. To recap: we (1) wrote a completely explicit formula for the area of a right triangle, (2) equated formulas corresponding to equal areas, and (3) found that the bulk of the formulas, everything except the part corresponding to the Pythagorean theorem, went away. The key to understanding all that is this picture:

It shows the area of a right triangle embedded in the area of the corresponding square, with the hypotenuse matching one side of the square. The precise formula for the ratio of the areas, , doesn’t matter so much — what matters is that when we blow this picture up or down, the ratio of the areas doesn’t change. That’s scale invariance. If the two smaller triangles add up to the big triangle, then the squares corresponding to the smaller triangles have to add up to the square corresponding to the big triangle. And that’s exactly the Pythagorean theorem.

I like to think of a2, b2, and c2 as units of area corresponding to each triangle. In a nutshell, the Pythagorean theorem decomposes a right triangle into two smaller, similar ones, and says that if the triangles add up, the units of area have to add up too. It’s deep, it’s direct, and I’ll never forget it. How about you?